Answer :
To solve the inequality [tex]\( |x-2| + 3 > 17 \)[/tex], we can follow these steps:
1. Isolate the Absolute Value: Start by moving the constant term to the other side of the inequality.
[tex]\[
|x-2| + 3 > 17
\][/tex]
Subtract 3 from both sides:
[tex]\[
|x-2| > 14
\][/tex]
2. Consider the Two Cases for the Absolute Value: The expression [tex]\( |x-2| > 14 \)[/tex] implies two possible scenarios:
- Case 1: [tex]\( x - 2 > 14 \)[/tex]
- Solve for [tex]\( x \)[/tex] by adding 2 to both sides:
[tex]\[
x > 16
\][/tex]
- Case 2: [tex]\( x - 2 < -14 \)[/tex]
- Solve for [tex]\( x \)[/tex] by adding 2 to both sides:
[tex]\[
x < -12
\][/tex]
3. Combine the Solutions: Since the original inequality involves an absolute value, the solution is a combination of the two cases above, representing all values of [tex]\( x \)[/tex] that satisfy either condition.
[tex]\[
x < -12 \quad \text{or} \quad x > 16
\][/tex]
Thus, the solution to the inequality [tex]\( |x-2| + 3 > 17 \)[/tex] is [tex]\( x < -12 \)[/tex] or [tex]\( x > 16 \)[/tex].
1. Isolate the Absolute Value: Start by moving the constant term to the other side of the inequality.
[tex]\[
|x-2| + 3 > 17
\][/tex]
Subtract 3 from both sides:
[tex]\[
|x-2| > 14
\][/tex]
2. Consider the Two Cases for the Absolute Value: The expression [tex]\( |x-2| > 14 \)[/tex] implies two possible scenarios:
- Case 1: [tex]\( x - 2 > 14 \)[/tex]
- Solve for [tex]\( x \)[/tex] by adding 2 to both sides:
[tex]\[
x > 16
\][/tex]
- Case 2: [tex]\( x - 2 < -14 \)[/tex]
- Solve for [tex]\( x \)[/tex] by adding 2 to both sides:
[tex]\[
x < -12
\][/tex]
3. Combine the Solutions: Since the original inequality involves an absolute value, the solution is a combination of the two cases above, representing all values of [tex]\( x \)[/tex] that satisfy either condition.
[tex]\[
x < -12 \quad \text{or} \quad x > 16
\][/tex]
Thus, the solution to the inequality [tex]\( |x-2| + 3 > 17 \)[/tex] is [tex]\( x < -12 \)[/tex] or [tex]\( x > 16 \)[/tex].
